On the bounding of limit multipliers for combined...

on June 3, 2018http://rspa.royalsocietypublishing.org/Downloaded from

Proc. R. Soc. A (2010) 466, 493–514doi:10.1098/rspa.2009.0240

Published online 21 October 2009

On the bounding of limit multipliersfor combined loading

BY M. FRALDI*, L. NUNZIANTE, A. GESUALDO AND F. GUARRACINO

Dipartimento di Ingegneria Strutturale, Università di Napoli ‘Federico II’,via Claudio 21, 80125 Napoli, Italy

In the framework of classical plasticity, even when limit multipliers and collapsemechanisms associated with different loads independently acting on a solid or structureare known, not much can be inferred on the limit multiplier of the combined loading.Frame structures under the action of dead loads and seismic forces, soil–foundationinteraction problems, tunnels under a variety of loads, deepwater pipelines subject tobending and pressure constitute only a few selected examples for which some sort ofsuperposition rule, as well as bounding techniques, would be extremely useful. Thepresent paper introduces a set of theorems for bounding limit multipliers for combinedloads. In particular, ranging from a minimum knowledge about the critical state under aparticular loading to a reasonable guess of the kinematics of the problem under combinedloads, more and more refined bounds for the overall limit multiplier are derived. Theresults, to the best of the authors’ knowledge, are novel and a few examples showingtheir practical value are presented and discussed.

Keywords: limit analysis; load multipliers; combined loading

1. Introduction

An important aspect of structural analysis, especially for ultimate safetyassessment or design, consists in evaluating the maximum load that the structurecan sustain. This is at the heart of many structural codes, and the earthquakewhich has recently shaken the central regions of Italy, wrecking homes andcausing a considerable death toll, is just the most recent example of the utmostimportance of a reliable evaluation of the collapse load in many structuralengineering problems. On account of the uncertainty of many parameters, theuse of methods based on the classic limit analysis, which avoids computationallyexpensive and often uncertain time-stepping analyses, represents a useful andvaluable approach. The distinctive feature of classical limit analysis is thedetermination of the load factor or, in practice, of its upper and/or lower

*Author and address for correspondence: Centro di Ricerca Interdipartimentale sui Biomateriali(CRIB), Università di Napoli ‘Federico II’, Piazzale Tecchio 80, 80125 Napoli, Italy([email protected]).

Received 4 May 2009Accepted 21 September 2009 This journal is © 2009 The Royal Society493

mailto:[email protected]

http://rspa.royalsocietypublishing.org/

494 M. Fraldi et al.


bounds at which a critical event occurs, namely plastic collapse. Therefore,in the past years, the limit analysis research field has been the focus ofintensive research efforts. In general, the most up-to-date formulations arederived within an optimization problem framework, aiming to take advantageof the latest mathematical developments in nonlinear convex programmingalgorithms. Nevertheless, in spite of the rapid evolution in computer performance,determining accurate collapse load estimates can still represent a significantcomputational effort. Additionally, as it is the case in most plasticity problems,the principle of superposition of loads does not hold true in the framework ofclassical limit analysis. Such a principle is extremely useful in the treatmentof many practical engineering problems and has been widely exploited in thepast two centuries on the basis of linearized strain and constitutive laws andlinear equilibrium equations for the stress state. In fact, in the linear theory ofelasticity, the response of a body under the action of any given combination ofloads is known once the solutions of the particular load cases are available. Inthe classical theory of plasticity, owing to both non-uniqueness of the stressesin strains and nonlinearity of the incremental elastoplastic process until thecollapse, superposition cannot be applied. However, in engineering applications,a large number of situations pose the problem of evaluating the failure undercombined loads, and the cases where superposition cannot be applied are ofgreat importance. Although critical loads are often available for each loadingcondition separately, serious difficulties can be instead encountered in thecomputation of the combined critical loading, where combined loading can besimply interpreted as the occurrence of more than one independent exertion factor(Zyczkowski 1981).

Some results are available for shakedown problems (König 1987), which ensurethe stabilization of plastic deformations under several loading histories, butno information is provided about combined critical loading, when unlimitedplastic deformation takes place while the different loads remain constantin time, notwithstanding the knowledge of the critical loading for eachloading case.

A superposition procedure has been recently employed by Puzrin & Randolph(2001) with reference to particular yield criteria and in the framework of upperbound limit analysis. They investigated the implications of combining twokinematically admissible velocity fields and showed that, under certain conditions,the superposition of two different collapse mechanisms can be used as a means ofimproving the upper bound collapse load for a defined loading. Thus, they makereference to superposition in a different way from what is meant in the presentwork and, moreover, the applicability of their findings is limited by the strictassumptions at the base of the treatment.

Frame structures under the action of dead loads and seismic forces, soil–foundation interaction problems, tunnels under a variety of loads, deepwaterpipelines subject to bending and pressure constitute a few selected examplesfor which some sort of superposition rule would be extremely useful. In all thesecases, bounding techniques can be extremely helpful. From a purely engineeringstandpoint, lower bounds to the limit carrying capacity are generally morerelevant than upper bounds, since in many practical applications safety factorsare needed. However, upper bounds may be employed to estimate the inaccuracywith respect to the actual limit multiplier.

Proc. R. Soc. A (2010)


Limit multipliers for combined loading 495


This said, on the sole basis of the tools of the classical theory of limit analysisand making resort to a set of inequalities, the present paper introduces a set oftheorems for bounding limit multipliers for combined loads. These findings, tothe best of the authors’ knowledge, are novel and a few examples showing theirpractical value are presented and discussed.

2. Framework

(a) Perfectly plastic behaviour

The fundamental theory of plasticity and limit analysis was pointed out more than50 years ago (Hill 1950; Prager & Hodge 1951). According to its foundations, inthe present work, it is postulated that (Maugin 1992):

(i) a strain energy ϕ(εel) function rules the elastic behaviour;(ii) a convex yield domain Y exists in the Westergaard principal stress space

and is defined by the condition

F(σ) = 0 �⇒ Y = {σ| F(σ) ≤ 0}; and (2.1)

(iii) a normal flow rule

εpl = λ∂Φ

∂σ, (2.2)

where λ is a scalar parameter, holds true.

Also, the flow rule is assumed to be associated and defines the plastic potentialas Φ = F(σ ).

Equation (2.2) can also be stated in a variational form, on the basis of theHill–Mandel dissipation principle, i.e.

(σ − σ∗) : εpl ≥ 0, ∀σ∗ ∈ Y , (2.3)

and the Drucker–Ilyushin stability postulate—that is,∫t

t0σ : ε(τ )dτ ≥ 0 for any

strain cycle—can be obtained as a direct consequence of equations (2.2) and(2.3) (Ilyushin 1948; Drucker 1988).

On account of the small-strain hypothesis, the following decomposition formulais also assumed to hold true

ε = εel + εpl = sym [∇ ⊗ u] �⇒ ε = εel + εpl = sym [∇ ⊗ u], ∀x ∈ Ω, (2.4)

where Ω ⊆ R3 stands for the domain occupied by the elastoplastic body. ∂Ωt and

∂Ωu denote the boundary regions where tractions and displacements are assigned,respectively.

Summarizing, it is

ε = εel + εpl = D : σ + λ∂F

∂σ,

⎧⎨⎩D = C

−1

λ ≥ 0 if {F = 0, F = 0}λ = 0 if F < 0 or {F = 0, F < 0}

(2.5)

C being the stiffness tensor in the linearly elastic range.





(b) The Greenberg minimum principle and the theorems of limit analysis

In the limit analysis, a stress field s(x) that obeys the field and boundaryequilibrium equations is called statically admissible. The set of all the staticallyadmissible stress fields is called S .

The plastically admissible stress fields belong to the convex domain Y ,

P ≡ {s|s(x) ∈ Y , ∀x ∈ Ω}. (2.6)

The set Σ is defined as Σ ≡ P ∩ S .A displacement/velocity field is called kinematically admissible if it is

mathematically well behaved (e.g. continuous and piecewise continuouslydifferentiable) and obeys the external and internal constraints, if any. The setof kinematically admissible velocity fields is called K .

The set of plastically admissible velocity fields u∗ can be introduced as

C ={u∗

∣∣∣∣∫∂Ωt

t · u∗ dS > 0}, (2.7)

where t is the boundary tractions and, without loss of generality, the hypothesis ofthe absence of body forces is postulated. Then, the set V is defined as V ≡ C ∩ K .

The classical theorems of limit analysis provide upper and lower bounds onthe loads under which an elastic perfectly plastic body reaches a critical state,i.e. a state in which large increases in plastic deformation become possible withlittle, if any, increase in load (Lubliner 1990; Khan & Huang 1995). This state iscalled unrestricted plastic flow and the loading state at which it becomes possibleis called ultimate or limit loading. In a state of unrestricted plastic flow, elasticitymay be ignored and therefore a treatment based on rigid-plastic behaviour is validfor limit analysis theorems.

The state of unrestricted plastic flow can be found on the basis of the Greenbergminimum principle (Drucker et al. 1952; Washizu 1982) by minimizing the totaldissipation functional, D,

minu∗ D, D =

∫Ω

σ : ε∗ dV −∫∂Ωt

t · u∗ dS =∫Ω

φ∗(ε∗) dV − λ

∫∂Ωt

t0 · u∗ dS

(2.8)among the admissible velocity fields u∗ ∈ V · t0 is the applied loading and λ isthe load multiplier. Given that the actual dissipated plastic power φ(ε) is

φ(ε) = σ : ε = supσ∈Y

(σ : ε∗) = supσ∈Y

φ∗(ε∗); (2.9)

since

φ(ε)

∣∣∣∣∂φ

∂ ε= σ, (2.10)





the value of the limit loading, λt0—and therefore the load multiplier, λ—is givenby Rayleigh’s quotient

λ =∫

Ωφ(ε) dV∫

∂Ωtt0 · v dS

= infu∗

( ∫Ω

φ∗(ε∗) dV∫∂Ωt

t0 · u∗ dS

), (2.11)

where v represents the actual velocity field at the collapse.On this basis, the lower and the upper-bound theorems of limit analysis can

be established.

(i) Lower-bound theorem

Let λs be a load multiplier. If Σ = P ∩ S(λs) �= ∅, i.e. if at least one stress fieldσ∗ ∈ Σ exists, then the following inequality holds true

λs ≤ λ. (2.12)

Therefore, λs is a safety factor.

(ii) Upper-bound theorem

Let λk be a load multiplier. If a kinematically and plastically admissible velocityfield u∗ ∈ V = C ∩ K (λk) �= ∅ exists such that∫

Ω

φ[ε(u∗)] dV ≤ λk

∫∂Ωt

t0 · u∗ dS , (2.13)

then it is

λk ≥ λ. (2.14)

Therefore, λk is an overload factor.Formulae (2.12) and (2.14) provide upper and lower bounds for the actual

plastic multiplier λ,λs ≤ λ ≤ λk . (2.15)

In the following, a compact notation will be adopted, that is

(σ, ε)Ω ≡∫Ω

σ : ε dV , (t, u)∂Ωt ≡∫∂Ωt

t · u dS , (2.16)

and the subscript ‘(. . .)’, i.e.

σ(···), ε(···), t0(···), u(···), . . . , S(···), P(···), K(···), C(···), Σ(···), V(···), . . . , (2.17)

will indicate the specific case under consideration.Without loss of generality, the absence of body forces is postulated.

3. Limit multiplier bounds for combined loads

As stated before, the scope of the present work is to establish some theoremswhich can yield relevant bounds on the overall limit multiplier in case of combinedloading (figure 1).





load 1

(a) (b) (c)

load 2 combined loads

Figure 1. An elastic–plastic body subject to (a,b) simple and (c) combined loading.

In simple words, say that for an elastic perfectly plastic body two differentloading conditions are prescribed on ∂Ωt, t0

1 and t02. If the corresponding limit

multipliers λ1 and λ2 (λ1 ≤ λ2) are determined, then what can be inferred aboutthe limit multiplier λ+ corresponding to the combined loading t0

1 + t02?

A first theorem, which can directly provide an overload factor with minimalinformation about the mutual dissipation of the single loading cases, can beimmediately stated.

Theorem 3.1 (A sufficient condition for λ+ ≤ λ1 (or λ+ ≤ λ2)). If the ‘mutualdissipation’ (t0

2, u1)∂Ωt (or (t01, u2)∂Ωt) is non-negative, then the limit multiplier λ+

for the combined loading cannot exceed λ1 (or λ2), that is(t0

2, u1)∂Ωt ≥ 0 �⇒ λ+ ≤ λ1 or (t01, u2)∂Ωt ≥ 0 �⇒ λ+ ≤ λ2. (3.1)

It is worth proving only the first relationship in equation (3.1), given thatdemonstrating the second one follows exactly the same steps. By virtue of theprinciple of virtual power (PVP) (Maugin 1980), the limit multiplier of thecombined loading can be written as

λ+ = (σ+, ε1)Ω

(t01 + t0

2, u1)∂Ωt

, (3.2)

where u1 ∈ V1 is the velocity field at the critical state under the sole traction t01.

Since it is

(t02, u1)∂Ωt ≥ 0 �⇒ λ+ = (σ+, ε1)Ω

(t01 + t0

2, u1)∂Ωt

≤ (σ+, ε1)Ω

(t01, u1)∂Ωt

, (3.3)

once again by virtue of PVP, it is found that the following equation holds true(σ+, ε1)Ω

(t01, u1)∂Ωt

= λ1(σ+, ε1)Ω

(σ1, ε1)Ω

. (3.4)

The stability postulate equation (2.3) implies that

(σ1 − σ+, ε1)Ω ≥ 0 �⇒ 0 ≤ α = (σ+, ε1)Ω

(σ1, ε1)Ω

≤ 1, (3.5)

where the inequality α ≥ 0 depends on the fact that, being (σ1, ε1)Ω > 0 and, byhypothesis, (t2, u1)∂Ωt ≥ 0, then it is(σ+, ε1)Ω ≥ 0 in order to be λ+ > 0.





Hence, inequality (3.3) becomes

λ+ = (σ+, ε1)Ω

(t01 + t0

2, u1)∂Ωt

≤ (σ+, ε1)Ω

(t01, u1)∂Ωt

= αλ1 �⇒ λ+ ≤ λ1 (3.6)

and the theorem is proved.On the other hand, with the scope of pursuing safety factors, a whole set of

bounds for the combined loading can be proved on the basis of a basic set ofinequalities. In order to do so, some lemmas are first established.

Lemma 3.2 (A sufficient condition for λ+ ≥ λ1). Let u+ ∈ V+ be the velocityfield at the critical state, with t0

1 and t02 acting at the same time. Then it is

(t02, u+)∂Ωt ≤ 0 �⇒ λ+ ≥ λ1. (3.7)

In fact, first of all, since u+ ∈ V+ = C+ ∩ K+, it is (t01 + t0

2, u+)∂Ωt > 0 andequation (3.7) implies (t0

1, u+)∂Ωt > 0.The limit multiplier λ+ with reference to the actual velocity field at the critical

state isλ+ = (σ+, ε+)Ω

(t01 + t0

2, u+)∂Ωt

. (3.8)

Under the hypothesis (3.7), it is

{(t01, u+)∂Ωt > 0, (t0

2, u+)∂Ωt ≤ 0} �⇒ λ+ = (σ+, ε+)Ω

(t01 + t0

2, u+)∂Ωt

≥ (σ+, ε+)Ω

(t01, u+)∂Ωt

, (3.9)

and by virtue of the stability postulate, it follows

(σ+ − σ1, ε+)Ω ≥ 0 ⇒ λ+ = (σ+, ε+)Ω

(t01 + t0

2, u+)∂Ωt

≥ (σ+, ε+)Ω

(t01, u+)∂Ωt

≥ (σ1, ε+)Ω

(t01, u+)∂Ωt

= λ1,

(3.10)which proves the lemma.

Lemma 3.3 (A sufficient condition for λ+ ≥ λ2 ≥ λ1). Under the same conditionsof lemma 3.2, it is possible to prove that

(t01, u+)∂Ωt ≤ 0 �⇒ λ+ ≥ λ2 ≥ λ1, (3.11)

the line of reasoning being the same as before.Lemma 3.4 (A necessary condition for λ+ ≤ λ1). Let u+ ∈ V+ be the velocity at

the critical state with two different loadings, t01 and t0

2, acting on the elastoplasticbody. Said λ+ the limit multiplier for the combined loading, it is

λ+ ≤ λ1 ≤ λ2 �⇒ {(t01, u+)∂Ωt ≥ 0, (t0

2, u+)∂Ωt ≥ 0}. (3.12)

In fact, by virtue of the Greenberg minimum principle, it isλ+(t0

1 + t02, u+)∂Ωt = (σ+, ε+)Ω , (3.13)

and both sides of equation (3.13) are positive quantities. Therefore, on the basisof the hypothesis, it can be written

λ1(t01, u+)∂Ωt + λ1(t0

2, u+)∂Ωt ≥ λ+(t01 + t0

2, u+)∂Ωt = (σ+, ε+)Ω (3.14)and, given the stability postulate and the PVP, it results

λ1(t02, u+)∂Ωt ≥ (σ+, ε+)Ω − λ1(t0

1, u+)∂Ωt = (σ+ − σ1, ε+)Ω ≥ 0. (3.15)

Since the limit multiplier is λ1 > 0, it follows that(t0

2, u+)∂Ωt ≥ 0, (3.16)





which proves the first statement in equation (3.12). By applying the same lineof reasoning with λ2 in place of λ1, the second statement can be easily proved,that is,

(t01, u+)∂Ωt ≥ 0. (3.17)

On the ground of the previous lemmas, the anticipated theorems bounding thevalue of the limit multiplier for combined loading on the sole basis of the limitmultipliers of the component loads are now proved. These results can also beviewed as some sort of superposition theorems for the limit multipliers and, forthe maximum clarity of the presentation, two different theorems will be proved:the first for the case of two-component loadings and the second for the case ofn-component loadings.

Theorem 3.5 (A lower bound for the overall limit multiplier λ+ in the case oftwo loadings). Let u+ ∈ V+ be the velocity at the critical state with two differentloadings, t0

1 and t02, acting on the elastoplastic body. Said λ+ the limit multiplier

for the combined loading, it is

k ≡ λ2

λ1≥ 1 �⇒ λ+ ≥ (λ−1

1 + λ−12 )−1 = λ1

m, m = 1 + 1

k. (3.18)

In order to prove equation (3.18), it can be noticed that

k ≡ λ2

λ1≥ 1 �⇒ m = 1 + 1

k> 1. (3.19)

In fact, three different cases may occur:

(I) {(t01, u+) > 0, (t0

2, u+) ≤ 0},(II) {(t0

1, u+) ≤ 0, (t02, u+) > 0}

and (III) {(t01, u+) ≥ 0, (t0

2, u+) ≥ 0}

⎫⎪⎪⎬⎪⎪⎭ (3.20)

given that the case {(t01, u+) ≤ 0, (t0

2, u+) ≤ 0} is not acceptable, being in contrastwith the condition of plastically admissibility of the velocity at the criticalstate, that is u+ ∈ C+(t0

1 + t02). Therefore, by virtue of lemmas 3.2 and 3.3, i.e.

equations (3.7) and (3.11), both relationships (I) and (II) in equation (3.20) implythat λ+ ≥ λ1 and equation (3.19) holds true.

From the third of equations (3.20), the following topological property can bededuced (figure 2),

{(t01, u+) ≥ 0, (t0

2, u+) ≥ 0} �⇒ u+ ∈ {V1 ∩ V2} �= ∅ �⇒ V+ ∩ {V1 ∩ V2} �= ∅,

(3.21)

being V(··· ) = K(··· ) ∩ C(··· ) the set of kinematically and plastically admissiblevelocity fields u(··· ).

Thus, this relationship and the PVP allow us to state that

λk1 = (σ+, ε+)Ω

(t01, u+)∂Ωt

≥ (σ1, ε+)Ω

(t01, u+)∂Ωt

= λ1

and λk2 = (σ+, ε+)Ω

(t02, u+)∂Ωt

≥ (σ2, ε+)Ω

(t02, u+)∂Ωt

= λ2

⎫⎪⎪⎪⎬⎪⎪⎪⎭, (3.22)





V+

V1

u2 ∈ V2 -

V2 -

u2 ∈ V1-

u+ ∈

V+

∩ {V2 ∩V2} ≠ 0-

Figure 2. Topological properties of the kinematically admissible velocity fields sets. Dashed line, V1;dashed dotted line V2; solid line, V+.

where λk1 and λk2 are the limit multipliers corresponding to the loading conditionst01 and t0

2, respectively. It follows that

1λ1

= (t01, u+)∂Ωt

(σ1, ε+)Ω

≥ 1λk1

= (t01, u+)∂Ωt

(σ+, ε+)Ω

and1λ2

= (t02, u+)∂Ωt

(σ2, ε+)Ω

≥ 1λk2

= (t02, u+)∂Ωt

(σ+, ε+)Ω

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (3.23)

and, by adding inequalities (3.23), it is

1λ1

+ 1λ2

≥ (t01, u+)∂Ωt + (t0

2, u+)∂Ωt

(σ+, ε+)Ω

= 1λ+

. (3.24)

which proves the theorem, since

λ+ > (λ−11 + λ−1

2 )−1 = m−1λ1, (3.25)

being λ2 ≡ kλ1 ≥ λ1 and m = 1 + k−1 > 1.

Corollary 3.6 (A general lower bound for the overall limit multiplier λ+). Evenif no information is available on the value of λ2, that is, λ2|{λ2 ≥ λ1, λ2 ∈ R+}, thefollowing inequality always holds true

λ+ ≥ λ1

2. (3.26)

In fact, since m is bounded, equation (3.26) straightforwardly follows fromrelationships (3.25) and (3.19), i.e.

∀k ≡ λ2

λ1≥ 1 �⇒ m = (1 + k−1) ∈ ]1, 2] ⊂ R+. (3.27)





It is worth noticing that inequalities (3.25) and (3.26) can be interpreted asrelatives of Gershgorin circle theorem (Varga 2004). In fact, as the Gershgorintheorem may be used to bound the spectrum of a square matrix, inequalities(3.25) and (3.26) can be employed to obtain an estimate of the actual limitmultiplier in the presence of the combined loads and to establish a safety factorwith regard to the critical state. Indeed, as a consequence of equation (3.27), itis easy to prove, for example, that

λ+ ≥ m−1λ1 ⇒ m−1λ1 = λs+ ∈ Σ+ ≡ S+ ∩ P+ (3.28)

In other words, by virtue of equation (3.28), the combined loading

t+ = λs+t0+ = (m−1λ1)t0

+ (3.29)

can be considered safe.

Corollary 3.7 (A partition technique for obtaining lower bounds). The limitmultiplier λ for an elastoplastic body under a certain load t0 can be bounded bypartitioning t0 in a set of loading conditions such that

t0 = t(1)

1 + t(1)

2 = t(2)

1 + t(2)

2 = · · · = t(n)

1 + t(n)

2 , n ∈ N, (3.30)

and whose limit multipliers {λ(1)

1 ≤ λ(1)

2 ; λ(2)

1 ≤ λ(2)

2 ; . . . ; λ(n)

1 ≤ λ(n)

2 } are known.Indeed, the following inequality is valid

∀i ∈ {1, . . . , n}, ∃j ∈ {1, . . . , n}|{k ( j) = λ( j)2

λ( j)1

= maxi

{k (i)},

m( j) = (1 + 1k ( j)

) = mini

{m(i)}} �⇒ λ ≥ λ( j)1

m( j)(3.31)

Moreover, even if λ(i)2 is unknown, provided ∀i ∈ {1, . . . , n}, λ

(i)2 ≥ λ

(i)1 , it is

∀i ∈ {1, . . . , n}, ∃h ∈ {1, . . . , n}|λ(h)

1 = maxi

{λ(i)1 } �⇒ λ ≥ λ

(h)

1

2. (3.32)

In fact, by applying theorem 3.5 to any arbitrary partitioning of theloading t0(provided the limit multipliers of the component loadings are known),equation (3.31) is simply obtained by choosing the largest value among the lowerbounds for λ. Then inequality (3.32) holds true by virtue of corollary 3.6.

It is worth highlighting the value of corollary 3.7 in the assessment of thedegree of safety of an applied loading in all those cases when the solution ofthe elastoplastic problem cannot be easily pursued. In fact, many problemsin engineering are amenable to be partitioned in two or more simple loadingcases and the proposed result allows the evaluation of the limit response of anelastoplastic body under the action of any given combination of loads, once thelimit multipliers of the particular load cases are available.

Finally, the extension of theorem 3.5 and of its corollaries to the case of n ∈ N

different loading conditions is given.





Theorem 3.8 (A lower bound for the overall limit multiplier λ+ in the case ofn loadings). If an elastoplastic body is subject to n ∈ N different loadings on itsboundary ∂Ωt, say

t0+ = t0

1 + t02 + . . . + t0

n =n∑

i=1

t0i , (3.33)

whose limit multipliers are known and it is

λmin ≡ λ1 ≤ λ2 ≤ · · · ≤ λn , (3.34)

the following inequality holds true

∀i ∈ {1, . . . , n − 1}, ki+1 = λi+1

λmin≥ 1 �⇒ λ+ ≥ m−1λmin, m = 1 +

n−1∑i=1

k−1i+1. (3.35)

Moreover, even if the multipliers λi+1 ≥ λmin, i ∈ {1, . . . , n − 1} are unknown, thefollowing inequality is applicable

λ+ ≥ λmin

n. (3.36)

Given hypotheses (3.33) and (3.34), a partition of t0+ such that

t0+ = t0

> + t0< (3.37)

can be introduced, with

t0> =

p<n∑j=1

t0j , j |{∀j ∈ {1, . . . , p < n}, (t0

j , u+)∂Ωt > 0}

and t0< =

n−p∑r=1

t0r , r |{∀r ∈ {1, . . . , n − p}, (t0

r , u+)∂Ωt < 0}

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭. (3.38)

Even if both the limit multipliers λ> and λ<, corresponding to the loads t0> and

t0<, respectively, are unknown, the results by lemmas 3.2 and 3.3 can be applied

to writet0+ = t0

> + t0< ⇒ λ+ ≥ λ>. (3.39)

Since the loading t0> can be regarded as a combined loading case, it is possible to

apply theorem 3.5 with respect to its limit multiplier, λ>, and obtain

λ> ≥⎛⎝p<n∑

j=1

λ−1j

⎞⎠−1

. (3.40)

Moreover, being any limit multiplier a positive number by definition, the followinginequality must also be true⎛⎝p<n∑

j=1

λ−1j

⎞⎠−1

>

(n∑

i=1

λ−1i

)−1

, (3.41)





thus it is

λ+ >

(n∑

i=1

λ−1i

)−1

. (3.42)

By virtue of this result and of corollary 3.7, it is also

λ+ > n−1λmin (3.43)

and, on account of the equality ki = λ−1minλi , the theorem is proved.

It is the case of noticing that inequality (3.25) resembles the Reuss lower-boundformula for the stiffness of a linearly elastic n-phase composite (Maugin 1992),with the limit multipliers in place of the elastic moduli. As a consequence, itcan be inferred that in case the limit load multiplier for each individual loadingpattern is not exact but it is estimated using lower- or upper-bound limit loadtheorems, the resultant limit multiplier evaluated by means of the proposedprocedure will be affected by an additional degree of approximation. Trivially,if the individual load multipliers are upper bounds of the actual ones, inequality(3.25) will provide a combined load multiplier which cannot be guaranteed to bea safety factor. The opposite happens in case of lower bounds.

No correspondence can be found, finally, with the Voigt upper bound.

4. A summary of the procedures for bounding the limit multiplier in the caseof combined loadings

On the basis of the findings of the previous section, it is finally possible tosummarize all the procedures to bound the limit multiplier in the case of combinedloads, showing their practical value. The presentation is restricted, for the sakeof clarity, to the case of two different loadings but it can be directly extended tothe case of n loadings by virtue of theorem 3.8.

A first value of the safety factor can be immediately obtained from expressions(3.25) and (3.26), on the sole basis of the values of all or, at least one, of the limitmultipliers of the particular loading, i.e.

λ+ > (λ−11 + λ−1

2 )−1

and λ+ ≥ λ1

2

⎫⎬⎭. (4.1)

However, the additional knowledge of the kinematics at the critical state of theparticular loadings, straightforwardly provides, by virtue of theorem 3.1, thefollowing overload factors

(t02, u1)∂Ωt ≥ 0 �⇒ λ+ ≤ λ1

and (t01, u2)∂Ωt ≥ 0 �⇒ λ+ ≤ λ2

}. (4.2)

Furthermore, if some basic information is available or can be easily inferred aboutthe kinematics of the critical state under combined loads, then lemmas 3.2 and 3.3





Table 1. Safety and overload multipliers for the combined loading, λ+, on the basis of the lemmasand theorems presented in §3. Without loss of generality, it is assumed λ1 ≤ λ2.

overload(t0

2, u1) ≥ 0 (t01, u2) ≥ 0 (t0

2, u+) ≤ 0 (t01, u+) ≤ 0 λ1 λ2 safety factor factor

× × × × • × λ1/2 ×× × × × • • (λ−1

1 + λ−12 )−1 ×

• × × × • • (λ−11 + λ−1

2 )−1 λ1× • × × • • (λ−1

1 + λ−12 )−1 λ2

× • • × • • λ1 λ2× × • × • • λ1 ×× × × • • • λ2 ×

allow us to establish that

(t02, u+)∂Ωt ≤ 0 �⇒ λ+ ≥ λ1

and (t01, u+)∂Ωt ≤ 0 �⇒ λ+ ≥ λ2 ≥ λ1

}, (4.3)

which constitute safety factors. It is important to note that

(λ−11 + λ−1

2 )−1 ≤ λ1 ≤ λ2 (4.4)

and, as a consequence, if the safety factors in equation (4.3) are available, theyconstitute better ones than those from equation (4.1).

In other words, ranging from a minimum knowledge about the critical state ofthe particular loading to a reasonable guess of the kinematics of the problem undercombined loads, more and more refined bounds for the overall limit multiplier canbe derived by virtue of the presented results, as summarized in table 1.

5. Examples

A few examples showing the actual value of the proposed formulae in engineeringproblems are finally presented and discussed in this section. The examples arechosen in order to present quantitative estimates of the actual carrying capacityof common structural problems and also to depict the regions which set the limitsof the plastic interaction surfaces. The use of upper bounds to estimate the errorwith respect to the actual limit multiplier is also shown.

(a) Example 1: frame structure under the action of combined horizontal andvertical point loads

With reference to figure 3, the frame structure can be subject to two differentloading conditions, that is a horizontal force H 0, a vertical force V 0 and acombined loading given by both the previous forces exerted on the structureat the same time.

The positions of possible plastic hinges can be immediately locatedand, therefore, all the collapse mechanisms are known (Massonet & Save 1980).





βL

(a)

(b)

L/2 L/2 L/2 L/2 L/2 L/2

β β

qV

lVV 0lHH0 l+H0 l+V0

qV qH

qH (1+2 tan b)qV (1+2 tan b) q2 (1+2 tan b)

qHq+ q+

Figure 3. Frame structure under the action of vertical, horizontal and combined point loadsconditions (a) and relative critical mechanisms (b).

Thus, the following limit multipliers are obtained

λV = 4M (3 + 2 tan β)

V 0L(1 + 2 tan β), λH = 2M (2 + tan β)

H 0L

and λ+ = 4M (3 + 2 tan β)

[H 0 + V 0(1 + 2 tan β)]L , (5.1)

where M = σYW pl is the plastic moment, σY is the yield stress and W pl is theplastic section modulus of the cross section, which is assumed the same for allthe elements. Theorem 3.5 gives the following lower bound, λL, for the combinedloading limit multiplier, λ+,

λ+ ≥ λL = (λ−1V + λ−1

H )−1 �⇒ H 0 + V 0(1 + 2 tan β)

(3 + 2 tan β)

≤ 2H 0

(2 + tan β)+ V 0(1 + 2 tan β)

(3 + 2 tan β). (5.2)

Inequality (5.2) can be trivially verified since 4 + 3 tan β > 0, ∀β ∈ [0, π/2[.Making reference to the collapse mechanisms illustrated in figure 3, it is also

immediately recognized that the mutual dissipation, as defined by equation (3.1),is positive,

H 0 × uV = H 0 × ϑVL > 0 and V 0 × uH = V 0 × ϑHL2

tan β > 0, (5.3)

where uV and uH denote the velocity fields at the critical states associated withthe vertical and horizontal point loads, respectively, and ϑ is the correspondingangular velocity. As a consequence, application of theorem 3.1 leads to an upper





o

x2

M2

M1x1

2σY

σY

2σr

σY

M

a

b

ϕα

n

Figure 4. Elliptical cross section of a beam subject to biaxial bending.

bound, λU, for the combined limit multiplier, λ+, that is

λ+ ≤ λU ≡ min{λV, λH} ={λV, β| {tan β < A − B, tan β > A + B}λH, β| {A − B < tan β < A + B} (5.4)

where

A = H 0

V 0− 5

4and B =

√(H 0

V 0

)2

+ H 0

2V 0+ 9

16. (5.5)

(b) Example 2: limit-carrying capacity of an elliptical cross section subject tocombined bending

Figure 4 shows the elliptical cross section of a cylinder subject to combinedbending M = [M 0

1 , M 02 ]T · a and b ≥ a are the main semi-diameters along the

principal axes {x1, x2}.By assuming that, within the cross section, the evolution of the normal stress

from the purely elastic to the fully plastic state yields a bi-rectangular distributionof the yield stress σY, with a discontinuity around the neutral axis

x2 = tan α x1, tan α = b2

a2tan φ, tan ϕ = M 0

2

M 01

, (5.6)

the plastic bending can be easily computed by taking into account the staticmoments of the regions delimited by the neutral axis (Zyczkowski 1981). Indeed,after some algebraic calculations, it is easy to obtain

M1 = 43ab2σY, M2 = 4

3a2bσY and M+ = 4

3ab

√b4 + a4 tan2 α

b2 + a2 tan2 ασY. (5.7)





Thus, the corresponding plastic multipliers are defined as follows:

λ1 ≡ M1

M 01

= 4ab2σY

3M 01

, λ2 ≡ M2

M 02

= 4a2bσY

3 tan φM 01

and λ+ ≡ M+M 0+

= 4a2b2σY

3M 01

√a2 + b2 tan2 φ

. (5.8)

The interaction curve can be also obtained by equating M 0+ =√

M 021 + M 02

2 to

M+. This leads to write, in the space of the generalized stresses {M 01 , M 0

2 }, theequation

Φ(M 01 , M 0

2 ) = 1 − 16a4b4σ2Y

9(a2M 021 + b2M 02

2 ), (5.9)

which plots an ellipse.A lower bound, λL, for the limit multiplier of the combined loading, λ+, can

be obtained by virtue of theorem 3.5. Indeed, equation (3.25) allows us to writethe inequality

λ+ ≥ λL ≡ (λ−11 + λ−1

2 )−1. (5.10)

Then, substitution of equation (5.8) into equation (5.10) gives

λ+ ≥ λL �⇒ c√a2 + b2 tan2 φ

≥ ca + b tan φ

, c = 4a2b2σY

3M 01

, (5.11)

which is trivially satisfied if the physically manifest hypotheses M 01 > 0 and

tan ϕ > 0 are assumed to hold true.In order to obtain an upper bound, λU, reference can be made to theorem 3.1.

The mutual dissipation, as indicated in equation (3.1), is indeed equal to zerofor both the first and the second loading conditions, being the bending moments{M 0

1 , M 02 } coaxial with the principal axes {x1, x2} of the cross section. The mutual

dissipation (3.1) written in terms of generalized stresses (bending moments M 0i )

and generalized plastic strains (rate of plastic curvatures χj), gives

{(M 01 , χ2) = 0, (M 0

2 , χ1) = 0} �⇒ λ+ ≤ min{λ1, λ2}. (5.12)

Thus, the most accurate upper bound λU is given by

λ+ ≤ λU = min{λ1, λ2} =⎧⎨⎩λ1, 0 < tan ϕ <

ab

λ2, tan ϕ >ab

. (5.13)

(c) Example 3: thin-walled tube subject to torque and tension

A thin-walled tube, with mean radius R and thickness t, is subject to theaction of a torque M 0

t and an axial force, N 0. This case represents one of thebasic testing arrangements to investigate yield criteria (Johnson & Mellor 1973)and therefore it can be considered to constitute a ‘gold standard’ for testing thematerial response to combined stresses. A fundamental prerequisite is that theaxial and shear stresses can be considered constant along the cylinder axis, aswell as in the hoop and radial directions on account of the thickness of the wall.





R

t

o

o

N0

N0

x3

x1

x1

x2

x2

Mt0

Mt0

θ

P

t 0

t 0

t 0

t 0

s 0

s0

Figure 5. A thin-walled tube under the action of axial force and torque.

Figure 5 shows an element extracted from the wall at a point P · σ 033 = σ 0 is

the tensile stress induced by N 0 and σ 03ϑ = τ 0 is the shear stress induced by M 0

t ,with {r , ϑ , x3} being a cylindrical coordinate system where x3 is the cylinder axis.Within this framework, it is

τ 0 = M 0t

2πR2tand σ 0 = N 0

2πRt. (5.14)

When the stress field is characterized by the normal and shear components only,say {σ , τ }, the principal stresses at P are

σI = 12σ +

√(14σ 2 + τ 2

), σII = 1

2σ −

√(14σ 2 + τ 2

)and σIII = 0, (5.15)

where {xI, xII, xIII} are the principal axes of stress. Thus, σY the yield stress of thematerial according to the von Mises criterion

Φ = (σI − σII)2 + (σII − σIII)

2 + (σIII − σI)2 − 2σ 2

Y = 2(σ 2 + 3τ 2) − 2σ 2Y = 0,

(5.16)

the plastic domain for any combination of the loadings N 0 and M 0t is given by

an ellipse in the plane {σ/σY, τ/σY} (figure 6a), whose equation is(σ

σY

)2

+(√

3τ

σY

)2

= 1. (5.17)





o A'A

B'

B

C'

C31/

1

o

sII

sY

sY

sY

sY

sI

ss

st

s0/sY

t 0/sY

t /sY

s /sYy0

etpl

espl

(a)

(b)

Figure 6. (a) Interaction curve in the plane {σ , τ }; (b) von Mises yield locus in the {σI, σII} plane: theplastic strain rates associated to the stress fields yielded by the axial force and torque, respectively,are drawn.

Equation (5.17) and figure 6 immediately suggest that the limit multipliers{λσ , λτ } can be written as

λσ := OA′

OA≡ σY

σ 0and λτ := OB

′

OB≡ σY√

3τ 0(5.18)

and, by following the same line of reasoning, the limit multiplier for the combinedloading is

λ+ := OC′

OC≡ 1√

1 + 3ψ20

σY

σ 0and ψ0 = tan ψ0 = τ 0

σ 0≥ 0. (5.19)





Theorem 3.5 states that a safety load multiplier λL can be found on account ofthe following inequality

λ+ ≥ λL = (λ−1σ + λ−1

τ )−1. (5.20)

Substitution of equations (5.16) and (5.17) in equation (5.20), after some algebraicmanipulations, allows to verify that

1√1 + 3ψ2

0

σY

σ 0≥ 1

1 + √3ψ0

σY

σ 0. (5.21)

In the range ψ0 ∈ [0, +∞] the per cent difference between the actual plasticmultiplier λ+ and its lower bound λL, that is,

e+L ≡ λ+ − λL

λ+= 1 −

√1 + 3ψ2

0

1 + √3ψ0

(5.22)

attains a maximum at ψ0 = 1/√

3, i.e. 30 per cent. Figure 7 shows the behaviourof the per cent error (5.22) over the range ψ0 ∈ [0, +∞[.

Finally, on account of theorem 3.5, an upper bound λU for the combined loadinglimit multiplier λ+ can also be obtained. In fact, since the stress is constant, thestrain rates at the critical state for both the simple loading cases are constant,too (figure 6b)

∀P ∈ Ω,

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩εσ ≡ εpl

σ = λσ

∂Φ

∂σ

∣∣∣∣σ=σσ

= 2σYλσ [2, −1, −1]T

ετ ≡ εplτ = λτ

∂Φ

∂σ

∣∣∣∣σ=στ

= 2√

3σYλτ [1, −1, 0]T, (5.23)

where σσ = [σI = σY, σII = 0, σIII = 0]T , στ = [σI = σY/√

3, σII = −σY/√

3, σIII =0]T .The parameters {λσ , λτ } have the meaning summarized in equation (2.5), and areobviously different from the limit multipliers.

By invoking the PVP, the sign of the external mutual dissipations can bededuced from that of the internal ones, that is,

(στ , εσ )Ω ≡ (στ , εplσ )Ω = 2

√3λσ σ 2

Y ≥ 0, (σσ , ετ )Ω ≡ (σσ , εplτ )Ω = 2

√3λτ σ

2Y ≥ 0.

(5.24)

Equation (5.24) points out that both the mutual dissipations are non-negative,so that, by virtue of equation (3.1), it has to be

λ+ ≤ λσ and λ+ ≤ λτ . (5.25)

On account of equations (5.18) and (5.19), the inequalities in equation (5.25) areeasily verified. On the other hand, the best choice for the upper bound, λU, of λ+





err

1.0

0.8

0.6

0.4

0.2

0.001

0.2

0.4

0.6

0.8

1.0

1.2

0.01 0.1 1 10 100 1000y0

l

lU

l+

lL

eL

eU

(a)

(b)

Figure 7. (a) Inaccuracy (%) of the overload (bold) and of the safety factor (dashed) with respectto the actual limit multiplier; (b) safety factor (dashed), overload (bold) and limit multiplier(continuous) for the combined loading.

depends on the ratio ψ0 = τ 0/σ 0. A study of equation (5.18) leads to

λ+ ≤ λU ≡ min{λσ , λτ } ={

λσ , 0 < ψ0 < 1/√

3λτ , ψ0 > 1/

√3

, (5.26)

and therefore

ΛL = 1

1 + √3ψ0

≤ Λ+ = 1√1 + 3ψ2

0

≤ ΛU ={

1, 0 < ψ0 < 1/√

31√3ψ0

, ψ0 > 1/√

3 , (5.27)

where the normalized multipliers Λ(··· ) = λ(··· )/λσ are employed to call off thecommon factor λσ = σY/σ 0. Figure 7 shows the effectiveness of the obtainedbounds.





Ly

o

M M

Ly

o

lk Fk0

D x = L /Nxk

Rk

xk

q0 q0

lk Fk – Rk

nl+ F0

l+ F0

Figure 8. A simply supported beam under the action of n forces, each spanned over alength �x = L/n.

(d) Example 4: simply supported beam under the action of n forces

This final example shows an elementary but significant application oftheorem 3.8.

Figure 8 shows a simply supported beam subject to an arbitrary but finitenumber of forces, such that

F 0k = F 0 = q0� x = q0 L

nk ∈ {1, 2, . . . , n}, n ∈ N. (5.28)

Each force, located at the position xk = L(k − 1/2)/n from the left end of thebeam, is affected by a certain multiplier, λk . The reaction at the support on theleft can be written as

Rk = λk(L − xk)

nq0 and R+ = λ+

q0

n

n∑k=1

(L − xk). (5.29)

Therefore, for the generic kth load case and for the combined loading, the limitmultipliers are, respectively,

λk = nMq0

× 1xk(L − xk)

and λ+ = 2nMq0L

× 1∑nk=1 (L − xk) − ∑(n−1)/2

k=1 (L − 2xk)= 8M

q0L2× n2

(1 + n2),

(5.30)

where M is the plastic bending. Without loss of generality, the analyticalexpression of λ+ has been obtained with reference to an arbitrary odd n, being aplastic hinge located at L/2. Therefore, by virtue of theorem 3.8, a lower bound,λL, can be easily obtained for an arbitrary number n of forces as

λ+ ≥ λL ≡ 1∑nk=1 1/λk

= 8Mq0L2

× 3n2

2(1 + 2n2). (5.31)





A comparison of the results found for λL and λ+ immediately points out thatinequality (5.31) always holds true, provided that n ≥ 1. Also, as can be expected,the per cent error e+

L can be estimated over the whole range of n and the followingresult is obtained

e+L = λ+ − λL

λ+= n2 − 1

2(2n2 + 1)�⇒ {

min e+L = e+

L |n=1 = 0, max e+L = e+

L |n→∞ = 1/4}.

(5.32)

6. Conclusions

The study has presented some theorems that yield relevant bounds on the overalllimit multiplier in the case of combined loads. The results can be regarded as asort of rule of superposition of the load multipliers in classical limit analysis.The findings appear of theoretical interest and have also been shown to beuseful in cases of combined loading. It has been pointed out that determiningupper and lower bounds to the combined loading multipliers may allow afairly accurate quantitative estimate of the actual carrying capacity of severalstructural problems.

References

Drucker, D. C. 1988 Conventional and unconventional plastic response and representation. Appl.Mech. Rev. 41, 151–167. (doi:10.1115/1.3151888)

Drucker, D. C., Prager, W. & Greenberg, H. J. 1952 Extended limit design theorems for continuousmedia. Q. Appl. Math. 9, 301–309.

Hill, R. 1950 The mathematical theory of plasticity. Oxford, UK: Clarendon Press.Ilyushin, A. A. 1948 Plasticity: elastic–plastic deformations (in Russian). Moscow, Russia:

Gostekhizdat.Johnson, W. & Mellor, P. B. 1973 Engineering plasticity. London, UK: van Nostrand Reinhold Co.Khan, A. S. & Huang, S. 1995 Continuum theory of plasticity. New York, NY: Wiley.König, J. A. 1987 Shakedown of elastic–plastic structures. Amsterdam, The Netherlands: Elsevier

and Polish Scientific Publishers.Lubliner, J. 1990 Plasticity theory. New York, NY: Macmillan Publishing.Massonet, C. & Save, M. 1980 Calcolo plastico a rottura delle costruzioni (in Italian). Milan, Italy:

Clup.Maugin, G. A. 1980 The principle of virtual power in continuum mechanics—applications to

coupled fields. Acta Mech. 35, 1–70. (doi:10.1007/BF01190057)Maugin, G. A. 1992 The thermomechanics of plasticity and fracture. New York, NY: Cambridge

University Press.Prager, W. & Hodge, P. G. 1951 Theory of perfectly plastic solids. New York, NY: John Wiley

and Sons.Puzrin, A. M. & Randolph, M. F. 2001 On the superposition of plastically dissipated work in upper

bound limit analysis. Proc. R. Soc. Lond. A 457, 567–586. (doi:10.1098/rspa.2000.0682)Varga, R. S. 2004 Geršgorin and his circles. Berlin, Germany: Springer-Verlag.Washizu, K. 1982 Variational methods in elasticity and plasticity, 3rd edn. Oxford, UK: Pergamon

Press.Zyczkowski, M. 1981 Combined loadings in theory of plasticity. Warsaw, Poland: Polish Scientific

Publishers.


http://dx.doi.org/doi:10.1115/1.3151888

http://dx.doi.org/doi:10.1007/BF01190057

http://dx.doi.org/doi:10.1098/rspa.2000.0682


On the bounding of limit multipliers for combined...

Documents

Transcript of On the bounding of limit multipliers for combined...